A new artificial-intelligence method designed to make radial basis function neural networks more accurate, adaptable, and stable could offer a fresh route toward modeling complex industrial systems. Researchers at the Beijing Institute of Petrochemical Technology have introduced a collaborative optimization algorithm that adjusts both the internal structure and numerical parameters of an RBF neural network according to how actively each neuron contributes to a prediction. The method, called AD-RBFNN, is based on a new measure known as activity density. According to the research team, this approach allows a network to identify which neurons are genuinely useful, refine their behavior, and remove or modify inefficient components while preserving the information needed for reliable modeling. The work addresses a long-standing challenge in neural-network design: finding the right balance between a model that is too simple to capture nonlinear behavior and one that is so complicated that it becomes unstable or performs poorly on new data.
RBF neural networks are particularly attractive for nonlinear system modeling because they can approximate complicated relationships using localized response functions. In a typical RBF network, the hidden layer contains neurons whose activation depends on the distance between an input and a neuron-specific center. A Gaussian radial basis function, for example, produces a strong response when an input lies near its center and a weaker response as the distance increases. The width of the function determines how broad that response is, while the output weights determine how strongly each neuron influences the final prediction. Together, the centers, widths, and weights shape the network’s representation of a system. However, these elements are closely connected. Changing the number of neurons can alter the balance of the entire model, while poorly selected centers or widths can cause underfitting, overfitting, slow training, or weak generalization.
The researchers’ central idea is to evaluate each neuron through its activity density, a quantity intended to describe the neuron’s average contribution relative to its width across the sample space. In practical terms, the measure considers how strongly a neuron responds and how broadly that response is distributed. A neuron with a large output but an excessively broad activation region may behave very differently from one that responds intensely within a narrow region. By combining response magnitude with width, activity density provides a way to distinguish neurons that make meaningful contributions from those that are redundant, weak, or poorly positioned. Rather than treating every hidden-layer unit as equally important, AD-RBFNN uses the activity profile of the network to guide structural and parameter changes.
This distinction is technically important because many conventional optimization strategies tune neural-network parameters without simultaneously reconsidering the network’s architecture. Other approaches may optimize the number of hidden neurons, but do so through a separate process that is not directly linked to the behavior of individual units. The proposed method attempts to make these decisions collaboratively. Activity density is used to guide changes to the centers, widths, output weights, and potentially the number of neurons. The aim is not simply to minimize training error, but to create a network whose internal structure reflects the distribution and complexity of the data. In a nonlinear system, different regions of the input space can behave in dramatically different ways. A network that allocates its representational capacity according to local activity may therefore be better equipped to capture sharp transitions, changing trends, and irregular process behavior.
The approach also includes a compensation mechanism intended to address a difficult consequence of structural adjustment. When neurons are added, removed, or modified, the output of the network can change abruptly, even if the new structure is theoretically more efficient. Such changes may introduce temporary errors or disrupt a model that was already performing well. The compensation mechanism is designed to offset the prediction disturbance caused by structural tuning. Conceptually, it allows the optimized network to preserve or recover essential output behavior while its internal organization is being revised. This is a crucial feature for applications in which the model must remain reliable during adaptation rather than being rebuilt from the beginning after every architectural change.
The researchers further provide an error-boundedness analysis to examine whether the method remains stable as optimization proceeds. Stability is especially significant in industrial modeling, where an inaccurate or rapidly changing prediction can affect process control, resource use, product quality, or environmental compliance. An algorithm may achieve a low error on a test dataset and still be unsuitable for deployment if its output can become unbounded or highly sensitive to small variations in input. By analyzing the behavior of the modeling error, the study seeks to establish theoretical conditions under which AD-RBFNN remains controlled. The error-boundedness result does not eliminate the need for appropriate data, parameter settings, and validation, but it provides a mathematical basis for arguing that the optimization procedure can be safely applied to nonlinear systems.
To test the method, the team applied AD-RBFNN to nonlinear function approximation, a standard benchmark for determining whether a model can reproduce complex input-output relationships. Such tests are useful because the underlying function can be known or precisely evaluated, allowing researchers to compare predictions against a clear reference. The algorithm was also used to model an industrial wastewater treatment process, an application where the relationship between operating conditions and treatment performance is influenced by multiple interacting variables. Wastewater systems can exhibit nonlinear dynamics because biological activity, chemical conditions, flow rates, pollutant concentrations, aeration, and environmental changes affect one another. A model that captures these relationships could help operators understand process behavior, anticipate changes, and make better-informed decisions about treatment conditions.
The reported experiments compared AD-RBFNN with other mainstream algorithms and found advantages in both model accuracy and generalization capability. The first measure reflects how closely the model reproduces the target behavior, while the second concerns how well it performs on data that were not used during training. This distinction matters because a highly flexible neural network can memorize the examples it has seen without learning the underlying process. By coordinating structural adjustment with parameter optimization, the activity-density method is intended to reduce this risk and produce a more useful representation of the system. The study’s findings suggest that the method can deliver more reliable predictions across the tested tasks, although the precise level of improvement depends on the dataset, comparison algorithms, training conditions, and evaluation criteria.
The work arrives as researchers and engineers seek neural-network techniques that are not only powerful but also easier to adapt and justify in real-world settings. Large deep-learning systems dominate many headline-making applications, yet smaller and more structured models remain valuable when data are limited, computation must be efficient, or the behavior of a system needs to be examined more directly. RBF networks offer a localized and mathematically interpretable architecture, and the activity-density concept adds a possible mechanism for making their internal organization more responsive to data. If validated across additional industrial processes, the method could support intelligent modeling in areas such as chemical production, energy systems, environmental monitoring, and process optimization. Its greatest significance may lie in treating neural-network architecture as a living component of the model—one that can be reshaped according to how information is actually used rather than fixed in advance.
The authors emphasize that AD-RBFNN is intended as a collaborative training framework in which network structure, parameters, compensation, and stability analysis work together. The study was conducted by Miao-Li Ma, Yan-Ni Su, Han Zhang, Xin-Yang Wang, Ran Dong, and Bin Wang of the Beijing Institute of Petrochemical Technology. The article was accepted for publication in Neural Processing Letters and made available as an early peer-reviewed version, with a permanent DOI and an open-access license that permits noncommercial sharing without creating derivative versions. Supported by funding from the National Natural Science Foundation of China, the Zhiyuan Science Foundation of BIPT, and the Beijing Municipal Education Commission, the research presents activity density as more than a new performance score: it is proposed as a guiding principle for constructing neural networks that can adjust themselves while maintaining bounded error. As industrial systems grow more interconnected and difficult to describe with traditional equations alone, such adaptive modeling strategies could become an important part of the next generation of intelligent process technologies.
Subject of Research: Collaborative optimization of radial basis function neural networks for nonlinear system modeling and industrial wastewater treatment prediction.
Article Title: A Collaborative Optimization Algorithm for RBF Neural Network Based on Activity Density
Article References: Ma, M.-L., Su, Y.-N., Zhang, H. et al. “A Collaborative Optimization Algorithm for RBF Neural Network Based on Activity Density.” Neural Processing Letters (2026).
Image Credits: AI Generated
DOI: 10.1007/s11063-026-11877-8
Keywords: Activity density; collaborative optimization; nonlinear system modeling; RBF neural network.
